3.49 \(\int \frac{\sin ^7(c+d x)}{(a-a \sin ^2(c+d x))^2} \, dx\)

Optimal. Leaf size=65 \[ \frac{\cos ^3(c+d x)}{3 a^2 d}-\frac{3 \cos (c+d x)}{a^2 d}+\frac{\sec ^3(c+d x)}{3 a^2 d}-\frac{3 \sec (c+d x)}{a^2 d} \]

[Out]

(-3*Cos[c + d*x])/(a^2*d) + Cos[c + d*x]^3/(3*a^2*d) - (3*Sec[c + d*x])/(a^2*d) + Sec[c + d*x]^3/(3*a^2*d)

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Rubi [A]  time = 0.0837538, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 2590, 270} \[ \frac{\cos ^3(c+d x)}{3 a^2 d}-\frac{3 \cos (c+d x)}{a^2 d}+\frac{\sec ^3(c+d x)}{3 a^2 d}-\frac{3 \sec (c+d x)}{a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^7/(a - a*Sin[c + d*x]^2)^2,x]

[Out]

(-3*Cos[c + d*x])/(a^2*d) + Cos[c + d*x]^3/(3*a^2*d) - (3*Sec[c + d*x])/(a^2*d) + Sec[c + d*x]^3/(3*a^2*d)

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

Rule 2590

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\sin ^7(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx &=\frac{\int \sin ^3(c+d x) \tan ^4(c+d x) \, dx}{a^2}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{x^4} \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (3+\frac{1}{x^4}-\frac{3}{x^2}-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac{3 \cos (c+d x)}{a^2 d}+\frac{\cos ^3(c+d x)}{3 a^2 d}-\frac{3 \sec (c+d x)}{a^2 d}+\frac{\sec ^3(c+d x)}{3 a^2 d}\\ \end{align*}

Mathematica [A]  time = 0.0504638, size = 59, normalized size = 0.91 \[ \frac{-\frac{11 \cos (c+d x)}{4 d}+\frac{\cos (3 (c+d x))}{12 d}+\frac{\sec ^3(c+d x)}{3 d}-\frac{3 \sec (c+d x)}{d}}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^7/(a - a*Sin[c + d*x]^2)^2,x]

[Out]

((-11*Cos[c + d*x])/(4*d) + Cos[3*(c + d*x)]/(12*d) - (3*Sec[c + d*x])/d + Sec[c + d*x]^3/(3*d))/a^2

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Maple [A]  time = 0.043, size = 47, normalized size = 0.7 \begin{align*}{\frac{1}{{a}^{2}d} \left ({\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3}}-3\,\cos \left ( dx+c \right ) -3\, \left ( \cos \left ( dx+c \right ) \right ) ^{-1}+{\frac{1}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^7/(a-sin(d*x+c)^2*a)^2,x)

[Out]

1/d/a^2*(1/3*cos(d*x+c)^3-3*cos(d*x+c)-3/cos(d*x+c)+1/3/cos(d*x+c)^3)

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Maxima [A]  time = 0.945454, size = 70, normalized size = 1.08 \begin{align*} \frac{\frac{\cos \left (d x + c\right )^{3} - 9 \, \cos \left (d x + c\right )}{a^{2}} - \frac{9 \, \cos \left (d x + c\right )^{2} - 1}{a^{2} \cos \left (d x + c\right )^{3}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^7/(a-a*sin(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/3*((cos(d*x + c)^3 - 9*cos(d*x + c))/a^2 - (9*cos(d*x + c)^2 - 1)/(a^2*cos(d*x + c)^3))/d

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Fricas [A]  time = 1.62859, size = 117, normalized size = 1.8 \begin{align*} \frac{\cos \left (d x + c\right )^{6} - 9 \, \cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} + 1}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^7/(a-a*sin(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

1/3*(cos(d*x + c)^6 - 9*cos(d*x + c)^4 - 9*cos(d*x + c)^2 + 1)/(a^2*d*cos(d*x + c)^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**7/(a-a*sin(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.16131, size = 77, normalized size = 1.18 \begin{align*} -\frac{32 \,{\left (\frac{3 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}}{3 \, a^{2} d{\left (\frac{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^7/(a-a*sin(d*x+c)^2)^2,x, algorithm="giac")

[Out]

-32/3*(3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 1)/(a^2*d*((cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 1)
^3)